We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper n-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph G satisfies the connected existentially closed property and admits a homomorphism to H, then it admits continuum-many distinguishing homomorphisms from G to H join K2. Applications are given to a family universal H-colourable graphs, for H a finite core.
Bonato et al. (Mon,) studied this question.
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